Problem: Find all integer values of $a$ so that the polynomial
\[x^3 + 3x^2 + ax + 7 = 0\]has at least one integer root.  Enter all possible values of $a,$ separated by commas.
Explanation: By the Integer Root Theorem, any integer root must divide 7.  Thus, the possible values of the integer root are 1, 7, $-1,$ and $-7.$

We can plug in each integer root separately to see what $a$ is in each case.  For $x = 1,$
\[1 + 3 + a + 7 = 0,\]so $a = -11.$  For $x = 7,$ $a = -71.$  For $x = -1,$ $a = 9.$  For $x = -7,$ $a = -27.$

Thus, the possible values of $a$ are $\boxed{-71, -27, -11, 9}.$